Spectral Analysis of Stochastic Neural Oscillators

大脑功能的许多方面涉及数百万个神经细胞的协调活动。单独来看，这些细胞可以像微小的时钟一样运作，发出稳定的相互通信的脉冲流。这些振荡器(“时钟”)的同步在健康和疾病的大脑状态中都起着作用。加深对脑细胞群如何同步(“滴答”在一起)或去同步化(“滴答”分开)的理解将加深对运动控制、癫痫、呼吸、信息处理和认知的大脑系统的理解。这个项目解决了对振荡神经细胞的理论理解中的一个概念空白。今天，大多数关于神经细胞如何同步的理论都依赖于这样的假设，即每个细胞的行为几乎是无懈可击的。然而，真正的神经细胞具有随机变异性，它们的行为在一定程度上是不规则的和不可预测的。在构建能够解释变异性的神经细胞的数学模型时，出现了具有数学挑战性的问题。解决这些数学问题可以提高定量描述单个神经细胞的钟状行为的能力。这个项目将通过提供对许多神经系统障碍的更深层次的了解来为大脑倡议做出贡献。这个项目解决了计算神经科学和神经生理学基础上的一个数学问题：在噪声存在的情况下，将振荡系统简化为相位振荡器的描述。具有稳定极限环的确定性动力系统的渐近相位是20世纪70年代在数学生物学中引入的。今天，相位振荡器模型的生物学意义怎么强调都不为过。相位和相位重置的基本概念依赖于确定性微分方程组的不变流形理论的经典结果。但噪声在生物动力学中是无处不在的。在重塑生物振荡器的模型以考虑随机波动时，经典的“渐近阶段”不再被很好地定义。该项目提出了一种新的噪声振子渐近相位的定义，用前向Kolmogorov算子的复本征函数来定义，描述了状态空间上密度的演化。在噪声强度为零的情况下，这个“随机渐近相位”与经典相位重合。然而，这一新定义并不依赖于相关确定性系统的经典相。与经典相位不同，对于需要噪声才能持续振荡的系统，它同样定义得很好。

Many aspects of brain function involve the coordinated activity of millions of nerve cells. Individually, these cells can behave like tiny clocks, emitting a steady stream of pulses that communicate with each other. The synchronization of these oscillators ("clocks") plays a role in both healthy and diseased brain states. Deepening the understanding of how groups of brain cells synchronize ("tick" together) or desynchronize ("tick" separately) will deepen the understanding of the brain systems underlying motor control, epilepsy, breathing, information processing, and cognition. This project addresses a conceptual gap in the theoretical understanding of oscillating nerve cells. Today, most theories about how nerve cells synchronize rely on the assumption that each cell behaves with nearly impeccable precision. However, real nerve cells have stochastic variability, and their behavior is partly irregular and unpredictable. In constructing mathematical models of nerve cells that can account for variability, mathematically challenging problems arise. Solving these mathematical problems can improve the ability to quantitatively describe the clocklike behavior of individual nerve cells. This project will contribute to the BRAIN Initiative by providing deeper insight into many nervous system disorders.This project addresses a mathematical problem at the foundations of computational neuroscience and neurophysiology: the reduction of oscillatory systems to a phase oscillator description, in the presence of noise. The asymptotic phase of a deterministic dynamical system possessing a stable limit cycle was introduced in mathematical biology in the 1970s. Today, the biological significance of phase oscillator models is difficult to overstate. The fundamental notions of phase and phase resetting depend on classical results of invariant manifold theory for systems of deterministic differential equations. But noise is ubiquitous in biological dynamics. In recasting models of biological oscillators to take into account random fluctuations, the classical "asymptotic phase" is no longer well defined. This project develops a new definition for the asymptotic phase of a noisy oscillator, defined in terms of the complex eigenfunctions of the forward Kolmogorov operator, describing the evolution of the density on the state space. This "stochastic asymptotic phase" coincides with the classical phase in the case of vanishing noise intensities. However this new definition does not rely on the classical phase of a related deterministic system. Unlike the classical phase, it is equally well defined for systems that require noise for sustained oscillation.